Diameter Minimization by Shortcutting with Degree Constraints

TL;DR

This paper addresses the challenge of adding a limited number of edges to a graph to minimize its diameter while respecting degree constraints, proposing algorithms, theoretical bounds, and empirical evaluations.

Contribution

It introduces three algorithms for diameter minimization with degree constraints, analyzes a special matching case, and provides empirical comparisons on real networks.

Findings

Algorithms effectively reduce diameter within constraints.

Matching augmentation has inapproximability bounds.

Empirical results demonstrate practical performance.

Abstract

We consider the problem of adding a fixed number of new edges to an undirected graph in order to minimize the diameter of the augmented graph, and under the constraint that the number of edges added for each vertex is bounded by an integer. The problem is motivated by network-design applications, where we want to minimize the worst case communication in the network without excessively increasing the degree of any single vertex, so as to avoid additional overload. We present three algorithms for this task, each with their own merits. The special case of a matching augmentation, when every vertex can be incident to at most one new edge, is of particular interest, for which we show an inapproximability result, and provide bounds on the smallest achievable diameter when these edges are added to a path. Finally, we empirically evaluate and compare our algorithms on several real-life networks of varying types.